3D THINKING ANIMATIONS
ANIMATION & ILLUSTRATION SUPPLEMENT TO “3D THINKING” BY CHAPTER, SUBJECT AND PAGE #
TITLE: “3D THINKING IN DESIGN AND ARCHITECTURE FROM ANTIQUITY TO THE FUTURE” 3D Thinking presents a history of the intimate relationships between geometry, culture, architecture and design throughout human history, from the Neolithic period through the Indian, Egyptian, Babylonian, Chinese, Greek, Celtic, Islamic, pre-Columbian and Renaissance cultures, to the present and the possible future. PUBLISHER: Thames & Hudson Ltd., April 2018. PRE-ORDER: Amazon USA, Amazon UK, Waterstones UK.
CHAPTER INDEX: Visual Logic 14. Neolithic Geometries 32. The River Cultures 58. The Americas 106. The Pythagoreans 124. European Tribal Geometry 160. Geometries of the Early Islamic Period 178. The Renaissance 206. Out of the Past – Into the Future 220. Modelling with Equations 222. Fractals 230. Shape-Changers 236 12. Dynamic Circles and Spheres 264. The Future of 3D Geometry 294. Additional Thoughts and Ideas 312.
VISUAL LOGIC
FROM SIGNS TO SYMBOLS – PAGE 15. The following shows the evolution of signs towards symbols from 70,000 to 5,000 years ago – where the European ice-age ended around 14,000 years ago. We see directional lines that criss-cross to form triangular shapes, shapes composed of directional lines that appear rectangular, linear arrangements of dots, ‘bell’ like enclosed shapes, feather like forms, comb like forms with various quantities of attached lines, a horse, spirals composed of dots, a composition of interconnected shapes, circular and curved shapes, curved triangles, a labyrinth, concentric circles (with carved bowls), symmetry spirals and lines. pattern repeats and spirals, zig-zag repeats, concentric arch-like line repeats.
THE MCGURK EFFECT – PAGE 22. An example of how our brain forms perceptions from fragmentary sensory information. Here, from a BBC 2 program, Prof Lawrence Rosenblum pronounces the word “ba” continuously. The video shows The professor pronouncing “ba” but then the video shows his mouth changing to pronounce the word “fa” – but the audio does not change. The effect is that our brain hears ‘fa” when the audio is still pronouncing “ba.” The effect was first discovered by Harry McGurk and John MacDonald at the University of Surrey.
NEOLITHIC GEOMETRIES
LABYRINTHS – ARCHITECTURE – PAGE 36 The Classic “Key Logic” labyrinth construction
LABYRINTHS – ARCHITECTURE – PAGES 37, 48. A Chartres Cathedral. The “switch” tile construction method
THE RIVER CULTURES
ANCIENT INDIA – THE INDUS VALLEY – THE VEDAS – PAGE 67 TO 72: The logic of Vedic Fire Altars. Additional to “3D Thinking” is the illustration of four fire altars constructed according to the traditions of the Vedas – all with the same surface area. See the post Fire Altars for accompanying text.
ANCIENT EGYPT – PYRAMIDS – ARCHITECTURE – PAGE 78: Of the two following animations one shows the comparative heights and slope angles of nine pyramids of ancient Egypt and the other shows a concept of a counter-balanced crane designed to lift large stone blocks*. The first graphic shows the comparative slopes and heights of the Giza pyramids. The second graphic shows how slope angles might have been controlled using timber frames and/or a central ‘sighting’ tower. *The concept of counter-balancing was known. Ratchets may have been used but there is no evidence of ratchets in surviving documents or in wall engravings or paintings. It would be possible to design a device that would function from the vertical position to an angle corresponding to the slope of the pyramid…there are lots of possibilities for counter-balancing.
ANCIENT CHINA – ICHING – PAGES 88 to 90: Fu Hsi’s He Tu (River Map): As legend has it Fu Hsi was fishing on the banks of the Yellow River some 5,000 years ago when a supernatural animal, a “Dragon Horse,” rose from the waters carrying on his back the river map hidden within which were the eight primary trigrams of the I-Ching. The following animation relates to the odd and even sequence on page 90 and a possible way by which Fu Hsi discovered the trigrams. The animation starts with the River Map then shows how the trigrams might have been derived followed by pathways of trigrams 0, 2, 4, 6 and then 1, 3, 5, 7 – where the pathways might be seen as a representation of the Dragon Horse. The last but one image in the animation shows the resulting order of trigrams and its symmetries – and the last shows the traditional arrangement.
THE PYTHAGOREANS
THE PYTHAGOREAN THEOREM – GEOMETRY – PAGE 131: There’s a visual balance to the animation shown below – a sort of harmony of proportion. The animation shows the Pythagorean Theorem expressed as circles or spheres where the sum of the two surface areas of the two inner circles/spheres always equals the surface area of the outer circle/sphere. The diameter of the outer circle/sphere equals the hypotenuse of a right-angled triangle. The diameters of the two inner circles/spheres equal the lengths of the other two sides of the right-angled triangle as its 90 degree vertex rotates around the circumference of the outer circle. The sum of the surface areas of the two inner circles/spheres always equals the surface area of the outer circle/sphere.
Pythagoras’ theorem states that the square of the hypotenuse of a right-angled triangle equals the sum of the squares of the other two sides. Whenever we’re considering areas of similar (same) shapes – in two or three dimensions – and where the sum of two areas equals the area of a third – then the dimensions of the three shapes will fall into the relationship of a^2 + b^2 = c^2. The reverse will also be true where any similar objects that have dimensions that fall into the relationship of a^2 + b^2 = c^2 will have areas that fall into area a + area b = area c. It’s all a matter of perspective.
GEOMETRIES OF EARLY ISLAM
THE “RAY” METHOD – ARCHITECTURE – PAGE 183: An Abbasid Caliphate method that serves as the basis for constructing surface designs with specific numeric values – often for ABJAD applications.
THE CLOSE-PACKING CIRCLE METHOD – ARCHITECTURE – PAGE 187: An Abbasid Caliphate method for efficiently organizing 2D space where connecting circle center and contact points creates lattices from which surface designs can be extracted. A development of this method is that of the “Dynamic Sphere Geometry.”
NESTING POLYGONS – ARCHITECTURE – PAGE 190: Nesting Octagons, see the blog ‘An Octagon Design System.’ Creates lattices from which surface designs can be extracted.
FRACTALS
THE MANDELBROT SET – PAGE 233: Black is within the set. Colors represent numbers close to being in the set. Credit lightscribe 1018389
SHAPE CHANGERS
MODULES 1, 12 – ARCHITECTURE – PAGE 236: An introduction the logic of Shape Changing Polyhedra can be seen in a paper I delivered to the Bridges Conference this August, 2016: http://archive.bridgesmathart.org/2016/bridges2016-225.pdf. Shape-Changing Polyhedra are three-dimensional forms composed of polygons that are flexibly connected. Of most interest are shape-changing polyhedra ‘shells’ that connect in a modular fashion to fill space – to fill space and still retain their shape-changing characteristics. The video-clips below show: (i) Core One shell,page 240. (ii) Two Core One shells combined, page 241, and five of eight equilibrium positions. (iii) Shows Core One modules connected in 3D space, pages 243 and 301.(iv) Core 12 in various shape-change positions, page 256.
DYNAMIC CIRCLES AND SPHERES
A FIRST SEQUENCE – PAGE 271: Animation of the First Sequence.
The first close-packing sequence, Fig 12.25 page 271, shown in the first animation below was based on a 4-circle rectangular arrangement where circles were allowed to incrementally increase and decrease in size and move along the lines of symmetry. Other limitations imposed for the sequence were that a central circle, r1, was always to stay in contact with one set of opposite lines of symmetry of the containing rectangle and that circles r3, and then r4 were given a priority of growth. There are seven close-packing arrangements in the sequence. What was of particular interest was that the circles in the second close-packing arrangement were in whole number relationships, corresponding with Soddy’s “Bowl of Integers,” and that the sixth arrangement generated, exactly, the “Golden Rectangle,” (√5+1) / 2 – a precise arrangement of three circle sizes of r1 = 2, r4 = 1, and r3 = (√5 – 1).
The second animation, Fig 12.44 page 280, shows how six equal size spheres in an equilateral triangular prism change size and position in small increments to transform into into a 3D development of the sixth arrangement with a sphere r5 is added to complete the close-packing.
The third arrangement, Fig 12.26 page 272, shows how the intriguing close packing arrangements will repeat across a 2D plane.
5-CIRCLE CLOSE-PACKINGS – PAGE 274: Fig 12.30 page 274 of the dynamic sphere geometry applied to 5-circles within the unit triangle of a square. The animation was generated by changing the size and position of the 5 circles in small incremental steps. The third close-packing in the animated sequence is that used to generate the Altair Design lattices and corresponds with the Abbasid Caliphate window, see Fig 7.54 page 187.
3D POLYHEDRA – ARCHITECTURE – PAGE 288: 3D line arrangements are generated by connecting sphere centers of the “Golden Ratio” sphere cluster HP1.3 – and polyhedra are abstracted from them. The animations shows an equilateral triangular base that indicates the position of repeats of the sphere polyhedra clusters.
PERCEPTUAL LINE PATTERNS – PAGE 293: A close-packing circle arrangement is used to create a line pattern by drawing non-convex polygons (star-polygons) at contact points of circles. The ‘star’ line pattern then serves as a “perceptual” line pattern from which an almost infinite number of images can be perceived (seen in the “minds eye.”).